Optimal. Leaf size=543 \[ \frac {3 \left (1-x^2\right )^{2/3} x}{8 \left (x^2+3\right )}-\frac {27 x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3}}-\frac {15 \tanh ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}+\frac {9\ 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {27 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{16 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3}}+\frac {5 \tanh ^{-1}(x)}{8\ 2^{2/3}} \]
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Rubi [A] time = 0.24, antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {470, 530, 235, 304, 219, 1879, 393} \[ \frac {3 \left (1-x^2\right )^{2/3} x}{8 \left (x^2+3\right )}-\frac {27 x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3}}-\frac {15 \tanh ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}+\frac {9\ 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {27 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{16 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3}}+\frac {5 \tanh ^{-1}(x)}{8\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 393
Rule 470
Rule 530
Rule 1879
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac {3 x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {1}{8} \int \frac {3-9 x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=\frac {3 x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {9}{8} \int \frac {1}{\sqrt [3]{1-x^2}} \, dx-\frac {15}{4} \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=\frac {3 x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3}}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3}}+\frac {5 \tanh ^{-1}(x)}{8\ 2^{2/3}}-\frac {15 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}-\frac {\left (27 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{16 x}\\ &=\frac {3 x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3}}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3}}+\frac {5 \tanh ^{-1}(x)}{8\ 2^{2/3}}-\frac {15 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}+\frac {\left (27 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{16 x}-\frac {\left (27 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{8 x}\\ &=\frac {3 x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {27 x}{8 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3}}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3}}+\frac {5 \tanh ^{-1}(x)}{8\ 2^{2/3}}-\frac {15 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}-\frac {27 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{16 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}+\frac {9\ 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{4 \sqrt {2} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 157, normalized size = 0.29 \[ \frac {1}{8} x \left (x^2 F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right )+\frac {3 \left (\frac {9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};x^2,-\frac {x^2}{3}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right )\right )-9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )}-x^2+1\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right )}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 4.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}} x^{4}}{x^{6} + 5 \, x^{4} + 3 \, x^{2} - 9}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4}{{\left (1-x^2\right )}^{1/3}\,{\left (x^2+3\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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